Cent. Eur. J. Phys. • 7(3) • 2009 • 432-443 DOI: 10.2478/s11534-008-0163-5

Central European Journal of Physics

Nonextensive perfect hydrodynamics – a model of dissipative relativistic hydrodynamics?

Research Article

Takeshi Osada1∗, Grzegorz Wilk2†

1 Theoretical Physics Lab., Faculty of Knowledge Engineering, Musashi Institute of Technology, Setagaya-ku, Tokyo 158-8557, Japan 2 The Andrzej Sołtan Institute for Nuclear Studies, Theoretical Physics Department, ul. Hoża 69, 00-681 Warsaw, Poland

Received 14 October 2008; accepted 19 December 2008

Abstract: We demonstrate that nonextensive perfect relativistic hydrodynamics (q-hydrodynamics) can serve as a model of the usual relativistic dissipative hydrodynamics (d-hydrodynamics) therefore facilitating consider- ably its applications. As an illustration, we show how using q-hydrodynamics one gets the q-dependent expressions for the dissipative current and the corresponding ratios of the bulk and shear viscosities to entropy density, ζ/s and η/s respectively. PACS (2008): 05.70.Ln, 12.38.Mh, 12.40.Ee Keywords: nonextensive • hydrodynamic models • high energy collisions • multiparticle produc- tion processes © Versita Warsaw and Springer-Verlag Berlin Heidelberg.

1. Introduction be very effectively described by using a thermodynamic approach in conjunction with hydrodynamical flow. In this so called Landau Hydrodynamic Model, the secondaries were considered to be a product of the decay of some Hydrodynamics is well established as an effective ap- kind of hadronic fluid, produced in such collisions, which proach to flow phenomena. Here we shall present its was expanding before hadronization [1–3]. This model has nonextensive relativistic version from the point of view of recently been updated to describe recent experimental high energy collision physics [1–14]. The characteristic data [4, 5]. Since then there have been a number of feature of such a processes is the production of a large successful attempts to develop new solutions for both the number of secondaries (multiplicities at present approach Landau model [6] and for the so called Hwa-Bjorken ver- ∼ 103). Already in 1953, when there were only ∼ 10 sion of the hydrodynamical model [7, 8], for which a new particles produced and registered, mostly in cosmic rays class of solutions has been found [9, 10]. Hydrodynamic experiments, it was found [1–3] that such processes can models of different types have therefore been frequently used in the phenomenological description of multiparticle ∗ production processes at high energies, especially for high E-mail: [email protected] † E-mail: [email protected] (Corresponding author) energy nuclear collisions [11–14]. These are of special

432 Takeshi Osada, Grzegorz Wilk

interest to us (being currently investigated at RHIC at fully clear (cf., [48])1. Brookhaven, with the newly commissioned LHC at CERN joining soon) because it is widely believed that in such collisions a new form of hadronic matter, the so called The physical picture emerging from the above experience Quark-Gluon-Plasma (QGP), will be produced [15–18]. is that, instead of a strict local thermal equilibrium customarily assumed in all applications of statistical models (including hydrodynamic), one rather encounters a kind of stationary state, which already includes some So far all hydrodynamic models in this field have been interactions. It can be introduced in different ways. based on the usual Boltzmann-Gibbs (BG) form of For example, in [54] it was a random distortion of the statistical mechanics. The only works discussing some energy and momentum conservation caused by the general features of the nonextensive hydrodynamics, surrounding system. This results in the emergence of which we are aware of [19–21], use a nonrelativistic ap- some nonextensive equilibrium. In [55–59] the two-body proach and are therefore not suitable for the applications energy composition is replaced by some generalized h E ;E we are interested in. On the other hand it is known energy sum ( 1 2), which is assumed to be associative that an approach based on non-extensive statistical but which is not necessarily simple addition and contains mechanics (used mainly in the form proposed by Tsallis contributions stemming from pair interaction (in the [22–28] with only one new parameter, the nonextensivity simplest case). It turns out that, under quite general parameter q) describes different sets of data in a better assumptions about the function h, the division of the way than the usual statistical models based on BG total energy among free particles can be done. Different statistics, cf., [22–28] for general examples and [29–47] forms of the function h then lead to different forms of the for applications to multi-particle production processes. entropy formula, among which one encounters the known Roughly speaking, all observed effects amount to a Tsallis entropy. The origin of this kind of thinking can broadening of the respective spectra of the observed be traced back to the analysis of the q-Hagedorn model secondaries (both in transverse momentum space and proposed some time ago in [60]. in rapidity space), they take the form of q-exponents instead of the naively expected usual exponents: 1/(1−q) exp(−X/T ) =⇒ expq(−X/T ) = [1 − (1 − q)X/T ] . All phenomenological applications of hydrodynamic From these studies emerged a commonly accepted models to the recent multiparticle production processes interpretation of the nonextensivity parameter q (in fact, show that, although perfect (nonviscous) hydrodynamics |q − 1|), as the measure of some intrinsic fluctuations successfully describes most RHIC data [6, 61], there are characteristic of the hadronizing systems under consid- indications that a hadronic fluid cannot be totally ideal. eration [29–31]. For q > 1, in transverse momentum For example, the perfect fluid dynamical calculations with space, q could represent fluctuation of the temperature, color glass condensate initial states could not reproduce T , corresponding to some specific heat parameter C. the elliptic flow data [63], indicating a necessity to use In this case q − 1 = C and therefore it should be some kind of viscous fluid description. A nonrelativistic inversely proportional to the volume of the interaction viscous fluid is usually described by the first order region. This effect is indeed observed [46, 47]. In rapidity space these are fluctuations of the so called 1 One should keep in mind, however, that there are du- partition temperature T E/hni , pt = [42–45], which are alities in the non-extensive approach, i.e., that both q precisely the same fluctuations that lead to the Negative and 1/q can be used as the nonextensivity parameter de- Binomial form of the observed multiplicity distributions, pending on the normalization of original or q-powered P(n; hni; k), with its characteristic parameter k given by probabilities. Also, when considering the particle-hole k = 1/(q − 1) [42–45, 48]. In the case of q < 1, the symmetry in the q-Fermi distribution, f(−E; T ; µ; q) = − f E;T; −µ; − q , in a plasma containing both par- interpretation is not at present clear [31]. It seems that 1 ( 2 ) ticles and antiparticles both q and − q occur (µ de- the first role of the parameter q is to restrict the allowed 2 notes the chemical potential here). These dual possibili- phase space [49]. Actually, the conjecture associating ties warn us that a theory requiring that only q > has q 1 with fluctuations has already been formalized as a physical meaning is still incomplete [22–28]. These points superstatistics new branch of statistical mechanics called deserve further considerations which are, however, going [50, 51]. It should be noted that there are also arguments outside the scope of this paper. On the other hand, in the connecting nonextensivity with some special dynamical present paper we are not considering plasmas containing correlations existing in the system under consideration both particles and antiparticles but only massive pions [52, 53], but their connection with fluctuations is not yet which are assumed to obey the q-Boltzmann distribution.

433 Nonextensive perfect hydrodynamics – a model of dissipative relativistic hydrodynamics?

Navier-Stokes equations. However, one needs their esis is now assumed in nonextensive form: special relativistic generalization and this turns out to be acausal and unstable [64, 65] (see also [66–71]). h f ; f f f ; q[ q q1] = expq [lnq q + lnq q1] One therefore looks towards the extended, second order theories accepting all problems connected with their for- where 1 1−q (1) mulation and practical applications [72–87]. Physically, expq(X) = [1 + (1 − q)X] ; the difference is that first order theories are based on (1−q) X − 1 the local equilibrium hypothesis, in which independent lnq(X) = : 1 − q variables are used, whereas in higher order theories only the fluxes of the local equilibrium theory appear as f x; p q independent variables. In particular, the entropy vector Here q( ) is the -version of the phase space distri- h f ; f q is quadratic in the fluxes, containing terms characterizing bution function, whereas q[ q q1] is the -version of the the deviation from local equilibrium. This situation, related to the presence of two par- x plus our experience with the nonextensive formalism, ticles in the same space-time position but with differ- p p [29–37, 42–48] prompted us to investigate the simple ent four-momenta and 1, respectively. By postulat- assuming nonextensive formulation of the perfect hydrodynamic ing Eq. (1) we are, in fact, that, instead of a a kind of stationary state is be- model, a perfect q-hydrodynamics [88–90]. It turned out strict (local) equilibrium, ing formed that this describes the experimental data fairly well. In , which already includes some interactions and q addition, an apparently unexpected feature appeared, which is summarily characterized by a parameter ; very namely the possibility that (relatively simple and first much in the spirit of [54–60] mentioned before. With this order) perfect q-hydrodynamics can serve as a model of assumption we are already departing from the picture of (second order and complicated in practical use) viscous the usual ideal fluid with its local thermal equilibrium, d-hydrodynamics. This is the point we would like to which is the prerogative of ideal hydrodynamics [64, 65]. discuss in more detail in this work. The consequences of this fact will be discussed below. The most important ingredient for further discussion is now the corresponding nonextensive entropy (q-entropy) current [88–91]: In the next Section we shall, for completeness, present the q Z d3p pµ n o main points of -hydrodynamics [88–90]. In Section3 we σ µ x −k fq f − f : q ( ) = B q lnq q q (2) propose a nonextensive/dissipative conjecture (NexDC), (2π~)3 p0 which allows us to connect ideal q-hydrodynamics with a d-hydrodynamics. Consequences of NexDC are discussed µ It turns out that ∂µσq ≥ 0 at any space-time point, i.e., in Section4 (entropy production) and in Section5 (trans- the relativistic local H-theorem is valid in this case [88– port coefficients). Section6 contains the summary 2 µ 92] . Assuming ∂µσq ≡ 0 one finds that (kB is Boltzmann constant):

µ 1/(1−q)  pµu x  f x; p − − q q( ) q( ) = 1 (1 ) k T x 2. Basic elements of B q( )  p uµ x  q−hydrodynamics µ q( ) ≡ expq − ; (3) kBTq(x)

µ where Tq(x) is the q-temperature [88–90] and uq(x) is the As in [88–90], we shall limit ourselves to a 1 + 1 dimen- q-hydrodynamic flow four-vector. Actually, one should be sional baryon-free version of hydrodynamic flow. This is derived following Lavagno [91], in which a nonextensive 2 Actually, this form of q-entropy current differs slightly version of the has been proposed and from that in [91]. The reason is that only with such form investigated. Because no external currents are assumed, we can, at the same time, both satisfy the H-theorem and this corresponds to a kind of perfect q-fluid. There are reproduce thermodynamical relations resulting in our q- two important points in [91]: (i) the Boltzmann equation , cf. Eq. (14), ((cf. also Eqs. (17a) and (17b) in q is formulated for the fq (x; p) distribution rather than for [88–90]), which are crucial in all further derivations and the usual fq(x; p); (ii) the usual molecular chaos hypoth- which were not addressed in [91]).

434 Takeshi Osada, Grzegorz Wilk

aware of the fact that there is still an ongoing discussion - (EoS) reflecting the internal on the meaning of temperature in nonextensive systems. dynamics of the fluid considered and (iii) freeze- However, the small values of the parameter q−1 deduced out conditions describing transformation of the ex- from data allow us to argue that, in the first approximation, panded fluid into observed hadrons. The problem Tq can be regarded as the hadronizing temperature in is that all of them can, in principle, enter with their such a system (cf., [93] for a thorough discussion of the own intrinsic fluctuation pattern, i.e., with their own temperature of nonextensive systems). Finally, we get the parameters q. In [88–90] where we provided prelim- q-version of the local energy-momentum conservation [91], inary comparison with experimental data, we have assumed, for simplicity, the same value of parame- µν ∂ν Tq (x) = 0; ter q throughout the whole collision. It is still to Z 3 µν 1 d p µ ν q be checked how good this assumption is. However, with Tq (x) ≡ p p fq (x; p); (2π~)3 p0 (4) in our present discussion this point is unimportant.  µ  pµuq(x) and fq(x; p) ≡ expq − : • Whereas in the usual perfect hydrodynamics (based kBTq(x) on the BG statistics) entropy is conserved in the hydrodynamic evolution, both locally and globally, In what follows we shall use covariant derivative notation µ µν in the nonextensive approach it is only conserved in which the vector u and tensor g are defined as locally. The total entropy of the whole expanding ν ν ν λ system is not conserved, because for any two vol- u ∂µu u V V ;µ = + Γλµ and V S( 1) S( 2) 6 umes of the fluid, 1;2, one finds that q + q = µν µν µ σν ν µσ (5) V ⊕V V g ∂µg g g ( 1 2) ( ) ;µ = + Γσµ + Γσµ Sq (where Sq are the corresponding total en- tropies). This should be always remembered (albeit, by means of the Christoffel symbols, strictly speaking, the hydrodynamic model requires only local, not global, entropy conservation). As ν 1 νσ Γλµ ≡ g (∂µgσλ + ∂λgσµ − ∂σ gλµ): a consequence of this fact, as we shall see below, 2 contrary to the situation in usual perfect hydrody- q In this notation Eq. (4) reads: namics, in the perfect -hydrodynamics the entropy is produced (but not q-entropy). µν  µ ν µν  Tq µ εq Pq uquq − Pqg ; = ( + ) ;µ • To guarantee that hydrodynamics makes sense, h i µ ν µν there should exist some spacial scale L such that = εq(Tq)uquq − Pq(Tq)∆q = 0; (6) ;µ the volume L3 contains enough particles. However, in the case when there are fluctuations and/or cor- µν ≡ gµν − uµ uν where ∆q q q. Here it was assumed that relations characterized by some typical correlation q T µν the -modified energy-momentum tensor q can be length l for which we expect that l > L, one has q (L3) decomposed in the usual way in terms of the -modified Sq ε ≡ u T µν u to use nonextensive entropy and its (locally energy density and pressure, q qµ q qν and (L3) s x Sq /L3 1 µν µ defined) density, q( ) = . When formulat- Pq ≡ − Tq qµν q uq 3 ∆ , by using the -modified flow (such µ ing the corresponding q-hydrodynamics one takes that in the rest frame of the fluid uq = (1; 0; 0; 0) and µ the limit L → 0, in which case the explicit depen- for q → 1 it becomes the usual hydrodynamic flow u ). dence on the scale L vanishes, whereas the cor- Notice that Eq. (6) is formally identical to the perfect relation length l leaves its imprint as a parameter hydrodynamic equation but with all the usual ingredients q. In this sense, perfect q-hydrodynamics can be replaced by their q-counterparts (perfect means here that considered as preserving causality and nonexten- there is nothing on the r.h.s. of Eq. (6)). In this sense we q perfect q-fluid sivity; is then related with the correlation length can speak about the mentioned before. l q ∼ l/L ≥ (one can argue that, very roughly, eff 1, L q where eff is some effective spacial scale of the - Some remarks are in order before proceeding further. hydrodynamics). If the correlation length l is com- L i.e. l ≈ L • When applied to multiparticle production processes, patible with the scale eff , , eff , one recovers each hydrodynamic model is supplemented with the condition of the usual local thermal equilibrium three ingredients which must be considered to- and in this case the q-hydrodynamics reduces to gether with Eq. (6) (or its equivalent): (i) - ini- the usual (BG) hydrodynamics. The above consid- tial conditions (IC) setting the initial energy den- erations were limited to the case of q > 1 only, for sity which is going to evolve hydrodynamically; (ii) which, as was said before, the clear correspondence

435 Nonextensive perfect hydrodynamics – a model of dissipative relativistic hydrodynamics?

with fluctuations was found. In what concerns the We tacitly assume that Tq is not too far from T (so far they case of q < 1, it seems that following [49], where are independent parameters but later on we shall impose q < 1 was found as a main factor closing the al- a condition on them, see Eq. (12) below). In Eq. (8) the µ lowed phase space, we can at the moment only pro- four-velocity u (x) is formally a solution of the equation pose that it could probably correspond to the case to which Eq. (6) is transformed by using Eq. (8): where, for some reason, the scale L cannot vanish l < L but must stop at some value . In this case, h µ ν µν µ ν µν i ε˜(Tq)u u − P˜(Tq)∆ + 2W ( u ) + π = 0: (9) analogously to what was said above, one could ex- ;µ q ∼ l/L pect that, again, eff , which this time would be smaller than unity. We shall not discuss this One can see that it has form of a dissipative hydro- µν µν µ ν possibility further in this work. dynamic equation [72–87] (here ∆ ≡ g − u u and def A(µBν) ≡ 1 AµBν Aν Bµ ε 2 ( + )) where ˜ is the energy density, µ P˜ is the pressure, W denotes the energy or heat flow 3. Nonextensive/dissipative conjec- µν vector, whereas π is the shear (symmetric and traceless) ture (NexDC) pressure tensor. They are defined as (using the angular

bracketdef notation: <µ ν>  1 µ ν µ ν 1 µν  λ σ q a b ≡ σ σ − λσ a b As seen in Eq. (6), the structure of the perfect - 2 (∆λ ∆ + ∆ ∆λ ) 3 ∆ ∆ ): hydrodynamical flow is formally identical with the flow of ideal fluid described by the usual ideal hydrodynamics. µ µ λ ε˜ = εq + 3Π; P˜ = Pq + Π;W = wq[1 + γ]∆λ δuq; We therefore regard the fluid described by Eq. (6) as a µ ν perfect q-fluid µν W W µν <µ ν> (in the sense already mentioned before). π wq δuq δuq ; = w γ 2 + Π∆ = Our experience with applications of q statistics [32–41] q[1 + ] tells us that the cases of interest to us are |q−1| << 1. It (10) is therefore tempting to simply expand the corresponding |q − | quantities in powers of 1 and to only keep the linear and expressed in terms of Π (being a kind of a q- term [60]. The result one gets looks promising, namely dependent bulk pressure), q-enthalpy wq and a q- dependent variable γ, T µν ≡ T µν q − τµν ; q q=1 + ( 1) q where g Z d3p  p · u  µν µ ν 1 2 Tq ≡ p p exp − ; and Π ≡ wq[γ + 2γ]; wq ≡ εq + Pq; =1 π 3 p0 T (7) (2 ) 3 (11) g Z d3p  p · u 2  p · u  µ 1 µ µν 1 µ ν γ ≡ uµδuq − δuqµδuq: τq ≡ p p exp − ; = 2 (2π)3 p0 T T 2

T q → q with q=1 being the usual energy-momentum tensor de- Notice that when 1, the difference between the µν δuµ → scribing an ideal fluid in the BG approach and τq repre- and ideal hydrodynamic flows vanishes, q 0, and γ → senting a viscous correction caused by the nonextensivity. with it also 0. This means that all dissipative fluxes d q However, in order for Eq. (7) to be valid in the whole phase of -hydrodynamics (9) which are induced by the -flow, p·u 2 W µ πµν space, the |1 − q| T < 2 inequality must hold. This like , and Π, vanish in this limit and one recovers means that either such a procedure can be applied only the equations of the usual perfect hydrodynamics. The γ to a limited domain of phase space, or that q = q(x; p), a variable is easier to handle and to calculate than the possibility which is outside the scope of the present work. differences in flows (see Eq. (17) below for its explicit We must therefore proceed in a more general way. Let form). Notice that, whereas the time evolution of Π is µ q us formally decompose εq, Pq and uq in Eq. (6) into the, controlled by -hydrodynamics (via the respective time ε P γ respectively, extensive and nonextensive parts: dependencies of q, q and ), its form is determined by the assumed constraints which must ensure that ε T ≡ ε T ε T ; the local entropy production is never negative (as is q( q) ( q) + ∆ q( q) nd always assumed in the standard 2 order theory [73–79] ). Pq(Tq) ≡ P(Tq) + ∆Pq(Tq); (8) uµ x ≡ uµ x δuµ x : q( ) ( ) + q( ) Now comes the crucial point. To finally link the usual q-hydrodynamics and its d counterpart, one has to fix Actually this can be only done approximately because our somehow the temperature Tq and the flow velocity field extensive ε and P still depend on Tq, rather than on T . uq. This is done by assuming that there exists such a

436 Takeshi Osada, Grzegorz Wilk

µ temperature T and velocity difference δuq that the fol- be divided into two parts: the energy-momentum tensor of T µν T µν q lowing two relations are satisfied: the usual ideal fluid, q=1 = eq , and the -dependent re- µν maining δT (the meaning of all components is the same δT µν → q → P(T ) = Pq(Tq); ε(T ) = εq(Tq) + 3Π: (12) as in Eqs. (10) and (11) and 0 when 1):

µν µ ν µν def µν µν Tq ≡ (εq + Pq)uquq − Pqg ≡ T + δT ; We call them the nonextensive/dissipative relations, eq NexDC in short. Here ε and P are the energy density where (16) i.e. µν µν (µ ν) µν and pressure as defined in the usual BG statistics, , δT = −Π∆ + 2W u + π : for q = 1. Eq. (12) provides a definite relation between Tq and T , therefore, in what follows, we shall mainly use Using now Eq. (12) we get from Eq. (14) that T for a description of dissipative effects (except of some p expressions when it is easier to keep both T and Tq, but γ = 1 + δq − 1; where always with the understanding that, because of Eq. (12) ε(T ) − εq(Tq) (17) δq ≡ : they are not independent). It is now straightforward to εq(Tq) + Pq(Tq) show [88–90] that in this case one can transform Eq. (9) into the equation of the usual d-hydrodynamics: This relation allows us to connect the velocity field uq (which is solution of the q-hydrodynamics given by Eq. (6)) µ ν µν with the velocity field u (which is solution of the dissipa- {ε(T )u u − [P(T ) + Π] ∆ tive hydrodynamics given by Eq. (13)). To make it more W (µuν) πµν : + 2 + ;µ = 0 (13) transparent, let us parameterize these velocity fields by using the respective fluid rapidities αq and α (the metric gµν ; − /τ2 This completes a demonstration of equivalence of the used is = (1 1 )): q perfect -hydrodynamics represented by Eq. (6) and its   d µ  1  -hydrodynamical counterpart represented by Eq. (13). uq(x) = cosh αq − η ; sinh αq − η ; q τ It means therefore that the perfect -fluid is nothing   (18) but a viscous fluid which satisfies the d-hydrodynamic uµ x α − η ; 1 α − η : ( ) = cosh ( ) τ sinh ( ) equation Eq. (13).

µ γ uµδuq αq − α − With the bulk pressure Π given by Eq. (11) and using Because = = cosh( ) 1 the connection u uq the NexDC relations (12) one can express the q-enthalpy between and is given by wq by the usual enthalpy, w ≡ T s = ε + P, and the α − α p δ q-dependent variable γ: cosh( q ) = 1 + q  p  =⇒ α = αq − log q + 1 + δq (19) ε(T ) + P(T ) wq εq Tq Pq Tq = ( ) + ( ) = γ 2 [1 + ] (we abandon the other solution, α = αq + w  p δ  = : (14) log q + 1 + q , because it leads to the reduction of γ 2 i.e. suµ < [1 + ] entropy, not to its production, , for it [ ];µ 0, for q > 1). Notice that Eq. (12) leads to the following important rela- µ tions between the heat flow vector W , the pressure tensor µν π and the bulk pressure Π: 4. Entropy production in q−hydrodynamics µ µν µ W Wµ = −3Πw; π Wν = −2ΠW ; µν (15) πµν π = 6Π2: One of the important implications of the NexDC conjecture is that in the ideal q-hydrodynamics one observes entropy As mentioned before, we expect that in all cases of interest production. Taking the covariant derivative of Eq. (16) and u to us |q−1| << 1. This means then that we can regard the multiplying it by ν one gets state characterized by fq(x; p) from Eq. (3) as some sta- µν µν near equilibrium near equi- uν T T suµ µ uν δT tionary state existing . This q;µ = [ ]; + ;µ = 0 librium state is then defined by the correlation function µ uν µν ⇒ su µ − δT : = [ ]; = ;µ (20) hq in Eq. (1), for which the energy momentum tensor can T

437 Nonextensive perfect hydrodynamics – a model of dissipative relativistic hydrodynamics?

in ε( ) = 22:3 GeV/fm3 (forcing the width of the q-gaussian to be equal to σ = 1:28), in the second the width of the q-gaussian was fixed as σ = 1:25 (forcing the initial en- in ergy density to be equal to ε( ) = 27:8 GeV/fm3). Because both (q-IC) give reasonable fits to experimental data, they introduce only very small differences in the entropy pro- duction, cf., the left part of Fig.1 . These (q-IC) were ac- companied by the q-dependent equation of state (q-EoS) for the relativistic pion gas. Finally, the usual Cooper- Frye prescription of the freeze-out (but with the corre- sponding distribution being the q-exponent rather than the usual one [88–90]) was used. All√ calculations were Au Au s performed for + collisions at NN = 200 GeV) (see [88–90] where it was shown that q-hydrodynamics with such q and with the respective q-IC and q-EoS re- dN/dy p produces and T distributions√ observed in RHIC s experiment for Au+Au collisions at NN = 200 GeV en- suµ > η τ ergy). Notice that ;µ 0 for large region at any (but especially for the early stage of the hydrodynami- cal evolution). It supports therefore a dissipative charac- ter of the q-hydrodynamics mentioned before and leads us to the conclusion that the equilibrium state generated in heavy-ion collisions may, in fact, be the q-equilibrium state, i.e., some stationary state near the usual equilib- rium state already containing some dissipative phenom- Figure 1. Illustration of different aspects of the entropy produc- ena. Notice that the total multiplicity, which is usually tion, suµ > . Two upper panels show production q [ ];µ 0 treated as a measure of entropy, increases with as ex- of entropy taking place in different parts of the longitu- N · q dinal phase space characterized by the rapidity variable, pected, namely = 11 (7 + 10 ). y = (1/2) ln[(t + z)/(t − z)], observed at different evolution moments τ. First panel shows results for the initial con- ditions given by fixed maximal initial energy density ε(in), second panel shows results for fixed width of the initial en- 5. Transport coefficients of ergy distribution (assumed in gaussian form) σ - see text for other details (cf., also [88–90]). Lower panel shows q−hydrodynamics how entropy is produced in the central region (for y = 0, first type of initial conditions were used here) for differ- ent values of the parameter q and for different evolution The other implication of the NexDC conjecture is that moments τ. This entropy production results in the cor- q responding growth with q of the multiplicity of produced there are some transport phenomena in an ideal -fluid. secondaries, N = 11 · (7 + 10q) (cf., Fig. 8 in [88–90]). We shall now identify the corresponding transport coef- ficients and compare them with the similar coefficients in the usual viscous d-hydrodynamics. Out of a number of different formulations of d-hydrodynamics [66–87] we q This means that, although in the ideal -hydrodynamics shall choose the 2nd order theory of dissipative fluids q conserved i.e. s uµ the -entropy is , , [ q q];µ = 0, the usual presented in [73–79]. It does not violate causality (at is produced q entropy ; the ideal -fluid is therefore a kind least not the global one over a distant scale given by the of usual viscous fluid. relaxation time) and it contains some dissipative fluxes This is illustrated in Fig.1 where the expected en- like heat conductivity and bulk and shear viscosities. We tropy production, as given by Eq. (20), is shown. All shall now see to what extent dissipative fluxes, resulting q curves presented in Fig.1 were calculated using the q- from our -hydrodynamics, can be identified with the hydrodynamical evolution described in [88–90] for q = heat conductivity and with the bulk and shear viscosities 1:08 with the q-dependent initial conditions (q-IC). They introduced in [73–79], and what are the resulting transport were given assuming a q-gaussian shape (in rapidity) of coefficients. the initial energy density out of which the hydrodynamic evolution started. Two types of (q-IC) were considered: in Let us start with the most general form of the off- µ the first the maximal initial energy density was fixed as equilibrium four-entropy current σ , which in our baryon-

438 Takeshi Osada, Grzegorz Wilk

free scenario takes the following form [73–79]: This is given by the second order polynomial in the bulk pressure Π. Therefore, it is natural to expect (and this will µ µ µν µ µν assumption σ = P(T )β + βν T + Q (δT ); be our ) that the most general entropy current uµ (21) in the NexDC approach is: βµ ≡ ; with T µ µ µ W µν µ Q su where δT is defined in Eq. (16). The function Q = full = Γ(Π) + Υ(Π) T Qµ δT µν µ ( ) characterizes the off-equilibrium state, which µ W = (χ + ξ)su + (χ − ξ) ; (26) in our case is induced by the nonextensive effects and T q Q → q → therefore depends on , 0 when 1. Using the where χ ≡ (Γ + Υ)/2 and ξ ≡ (Γ − Υ)/2 q NexDC conjecture, Eqs. (12), and Eq. (14) our -entropy (27) current is

µ ; µ µ µ W µ and where Γ Υ are (in general infinite) series in powers of σ ≡ s su Q ; µ q ( q) = + + q Q T the bulk pressure Π. In this sense full can be regarded as  W µ  (22) full order dissipative current q Qµ Qµ ≡ χ suµ : the in -hydrodynamics. In where q = χ + T i.e. σ µ 6 general one has entropy production/reduction, , ;µ = 0, however, in the case when Γ(Π) = Υ(Π) = χ one has σ µ ; Our near equilibrium state is thus characterizes by the χ;µ = 0. Out of the two possible solutions for (Γ Υ) only µ function Qχ in which one is acceptable,

r r ! T 3Π 1 T T γ T χ ≡ 1 − − 1 = · − 1 (23) − 3Π − − 2 ; Tq w γ Tq Γ = 2 1 1 = 1 + Tq w 1 + γ Tq (28) T − Tq (temperatures T and Tq are not independent but con- Υ = 2 ; Tq nected, as was stated before, by the NexDC conjecture (12)). We stress that it is given by the dissipative part of µ q uµQ ≤ i.e. our -system, the one that leads to the increase of the because only for it full 0 ( , the entropy is maxi- q usual entropy, cf. Fig.1 (the -entropy, as was discussed mal in the equilibrium [76–79], this is because (T −Tq)/Tq σ µ before, remains, however, strictly conserved, q;µ = 0). is always positive for q ≥ 1 [88–90]). In this way we finally arrive at the following expression for the full or- µ The most general algebraic form of Q for d-hydrodynamic der dissipative entropy current emerging from the NexDC that includes dissipative fluxes up to second order is [76– approach 3: 79]: " r #   µ −β 2 β W W ν − β π πνλ µ µ W 2T 3Π µ µ 0Π + 1 ν 2 νλ µ σ ≡ su − − − su Q u full + 1 1 = T Tq w 2nd 2T α W µ α πµν W µ 0Π 1 ν 2(T − Tq) W − + ; (24) + : (29) T T Tq T

β where i=0;1;2 are the corresponding thermodynamic co- efficients for the, respectively, scalar, vector and tensor 3 One must keep in mind therefore that, although to de- dissipative contributions to the entropy current, whereas fine the entropy current in q-statistics we require only α that q > , the NexDC correspondence requires in ad- i=0;1 are the corresponding viscous/heat coupling coeffi- 0 dition that q > 1 to be consistent with d-hydrodynamics cients. The corresponding expression calculated using the [76–79]. Only then, as witnessed by Fig.1 , constant NexDC conjecture is: (nonzero) initial Tsallis entropy results in increasing BG entropy (demonstrating itself in the total multiplicity in- W µ Qµ Qµ suµ ; creasing with q− ). For q < we could have decreasing q = =Γ2nd + Υ1st ( 1) 1 2nd T BG entropy from some positive initial value (and, accord- β β β ≡ − 3 1 − ( 0 + 6 2) 2; (25) ingly, total multiplicity decreasing with (q − 1)). This where Γ2nd Π Π 2 2w point is, however, not totally clear at present and we plan ≡ − α α : to address it elsewhere. and Υ1st ( 0 + 2 1)Π

439 Nonextensive perfect hydrodynamics – a model of dissipative relativistic hydrodynamics?

form:

µ µ µ σ χ µ ξ µ; full;µ = [(1 + )Φ ]; + [ Ψ ]; W µ µ suµ where Φ = + T (31) W µ µ suµ − : and Ψ = T

q σ µ Because conservation of the -entropy, q;µ = 0, is equiv- µ χ µ alent to [(1 + )Φ ]ν; = 0, therefore, using Eq. (15) one µ − W Wν uµ Wν πµν gets that Ψ = 3ΠT + 2ΠT , therefore

µ µν µ Π µ W π σ − wu Xµ − Y˜µ Zµν ; full;µ = T ( ) T + T (32)

where:   ξ ∂µΠ ∂µT ∂µξ Xµ = − + + ; Π Π T ξ ξ   Y 2 uν W 1 W uν − 1 πν ; µ = µ;ν + µ ν µ ν Π 3 3 ; 2 ; ξ   Z 1 W ; µν = ν;µ Π 2 Y˜µ = Yµ − ΠXµ;

Y˜µWν Figure 2. Illustration of the sum rule (34) for different values of tem- Z˜µν ≡ Zµν + : perature T and different nonextensivities q (the other pa- 2Π rameters are the same as in Fig.1 for the first type of initial conditions). Eq. (15) allows to eliminate the term proportional to the W µ heat flow, T . Finally one obtains

µν µ Π µ π T/T ≈ σ − wu Xµ Z˜µν Limiting ourselves to situations when q 1 and ne- full;µ = T ( ) + T O /w 2 glecting terms higher than (3Π ) , one obtains that 2 µν Π π πµν = + ≥ 0; (33) ζT 2ηT

"    2# where we have introduced the usual bulk and shear vis- µ 3Π 1 3Π µ Q ≈ − − su : cosities, ζ and η. Notice that, because of Eq. (15) one full w w (30) 4 avoids the explicit contribution to the entropy production W µ coming from the heat flow, T , which is present in Eq. (32) when one discusses a baryon free fluid, in which case the necessity to use the Landau frame would appear. As β 2 Comparing now Eqs. (25) and (30) one gets that 1 = w , one can see, Eq. (31) is covariant and therefore it does β β 9 α α 0 + 6 2 = 2w and 0 + 2 1 = 0. Since in the Israel- not depend on the frame used. One arrives at our main τ Stewart theory [73–75] the relaxation time is propor- result: the sum rule connecting bulk and shear viscos- β tional to the thermodynamic coefficients 0;1;2, it is natu- ity coefficients (expressed as their ratios over the entropy rally to assume that in the case of NexDC it is propor- density s), see Fig.2 : tional to the inverse of the enthalpy, τ ∝ 1/w (notice µ that for the classical Boltzmann gas of massless particles wσ µ β /w 1 3 full; : 2 = 3 [76–79, 87]). + = (34) ζ/s η/s Π2 Let us now see what kind of bulk and shear viscosities emerge from the NexDC approach. To this end let us write This is as much as one can get from the q-hydrodynamics the full order entropy current Eq. (26) in the following alone. To disentangle this sum rule, one has to add some

440 Takeshi Osada, Grzegorz Wilk

additional input. Suppose, therefore, that we are inter- ested in an extremal case, when a total entropy is gen- erated by action of the shear viscosity only. In this case one can rewrite the first part of Eq. (33) as

µν µ π h πµν λ i σ µ = − (wu Xλ) + Z˜µν ; (35) full; T 6Π

and arrive at

− " µν # 1 η γ(γ + 2) π Z˜µν λ = − su Xλ : (36) s (γ + 1)2 Π T

The predictions of Eq. (36) are shown in Fig.3 for different values of the parameter q and for different rapidities. They are also confronted with the known result on η/s, provided by the popular Ads/CFT conjecture [94–97], that η/s ≥ 1/4π. Assuming the validity of this limitation we can use Eq. (34) only in the region where the r.h.s. of Eq. (36) is smaller than (or equal to) 1/4π (in which case we put η/s = 1/4π), otherwise (because of our assumption that the total entropy is generated by the shear viscosity only) we have to put ζ/s = 0 and use Eq. (36) to evaluate η/s. The results for ζ/s and η/s are shown in Fig.3. Notice that, when the r.h.s of Eq. (36) approaches 1/4π, ζ/s given by Eq. (34) approaches infinity. To avoid such a situation, η/s should start to increase at higher temperatures, for example at T ≥ 75 MeV 4.

6. Summary

To summarize, we have discussed dissipative hydrodynam- ics from a novel point of view. This is provided by the

Figure 3. Upper and middle panels: the ratio of the shear viscosity 4 It should be kept in mind that we have so far obtained over the entropy density, η/s, as function of temperature only the relations between the Israel-Steward coefficients T , calculated using Eq. (36) for different values of q at the (β ; ; and α ; ) and not their individual values, there- mid-rapidity region y = 0 (two upper-left panels) and for 0 1 2 0 1 rapidity y = 3 (middle panels). The same version of hydro- fore we cannot compare our results to those of the ideal dynamical model was used as in Fig.1 with the first type Boltzmann gas case. This is due to the tensor form of of initial conditions. Notice that for q → 1 this ratio van- relations (15) and to the fact that the original perfect q- ishes, as expected, proving therefore correctness of our numerical calculations. The lower two panels summarize hydrodynamics (Eq. (4)) does not contain any natural the rapidity dependence of both the shear and the bulk space-time scale. It is then natural that NexDC conjec- viscosity ratios, η/s (left panel) and ξ/s (right panel). The ture does not introduce per se any definite relaxation time η/s is calculated using Eq. (36), i.e., assuming ζ/s = 0, or viscous-heat coupling length scale. Also, in examples and presented only above the limit value η/s = 1/4π found in the AdS/CFT approach [94–97]. The right lower panel shown here, only EoS for pionic gas was used as in [88– shows the bulk viscosity, ζ/s, calculated in this limit, i.e., 90]. Although it was shown there that such EoS depends assuming η/s = 1/4π and using Eq. (34). only very weakly on the parameter q, it remains to be checked whether this is also true for a more realistic EoS with quarks and gluons and in the vivinity of the QGP → hadronic matter . We plan to address this point elsewhere.

441 Nonextensive perfect hydrodynamics – a model of dissipative relativistic hydrodynamics?

nonextensive formulation of the usual perfect hydrodynam- ical model recently proposed by us [88–90]. Such a model References can be solved exactly and contains terms which can be interpreted as due to some dissipative effects. They can be identified and are expressed by the nonextensivity pa- [1] L. D. Landau, Izv. Akad. Nauk SSSR 17, 51 (1953) rameter q of the Tsallis formalism applied here. This find- [2] S. Z. Belenkij, L. D. Landau, Usp. Fiz. Nauk+ 56, 309 ing was used to propose a possible full order expression (1955) σ µ for the dissipative entropy current full resulting from the [3] S. Z. Belenkij, L. D. Landau, Nuovo Cim. Suppl. 3, 15 nonextensive approach. The corresponding bulk and shear (1956) transport coefficients resulting from q-hydrodynamics are [4] C-Y. Wong, Phys. Rev. C 78, 054902 (2008) connected by a kind of sum rule, Eq. (34). They were cal- [5] C-Y. Wong, arXiv:0809.0517 culated for some specific simplified case, cf. Fig.3 . We [6] L. M. Satarow, I. N. Mishustin, A. V. Merdeev, H. close by noticing that there is still some uncertainty in the Stöcker, Phys. Rev. C 75, 024903 (2007) equation of state used in our numerical example (for exam- [7] R. C. Hwa, Phys. Rev. D 10, 2260 (1974) ple, in what concerns the role of the possible QGP phase [8] J. D. Bjorken, Phys. Rev. D 27, 140 (1983) transition, which, when included in EoS, could consider- [9] T. Csörg˝o, M. J. Nagy, M. Csanád, Phys. Lett. B 663, ably affect η/s presented in Fig.3 ). We plan to address 306 (2008) this subject elsewhere. [10] M. J. Nagy, T. Csörg˝o, M. Csanád, Phys. Rev. C 77, 024908 (2008) From our point of view it is interesting to observe that [11] P. Huovinen, P. V. Ruuskanen, Ann. Rev. Nucl. Part. (at least some) remedies proposed to improve the formu- Phys. 56, 163 (2006) lation of d-hydrodynamic [64, 65, 70, 71] (like the use of [12] T. Hirano, J. Phys. G Nucl. Partic. 30, S845 (2004) some induced memory effects) introduce conditions which [13] T. Hirano, Y. Nara, J. Phys. G Nucl. Partic. 31, S1 in statistical physics lead in a natural way to its nonex- (2005) tensive version described by the parameter q [22–28]. It [14] Y. Hama, T. Kodama, O. Socolowski Jr., Braz. J. Phys. is then natural to expect that a nonextensive version of 35, 24 (2005) the hydrodynamical model with the nonextensivity param- [15] B. Müller, Nucl. Phys. A 774, 433 (2006) eter q could help us to circumvent (at least to some ex- [16] M. Gyulassy, L. McLerran, Nucl. Phys. A 750, 30 tent) the problems mentioned. This is due to the fact (2005) that equations of the perfect nonextensive hydrodynamics [17] I. Vitev, Int. J. Mod. Phys. A 20, 3777 (2005) (or perfect q-hydrodynamics) can be formally solved in an [18] R. D. Pisarski, Braz. J. Phys. 36, 122 (2006) analogous way as equations of the usual perfect hydrody- [19] B. M. Boghosian, Braz. J. Phys. 29, 91 (1999) namics [88–90]. However, it turns out that, from the point [20] F. Q. Potiguar, U. M. S. Costa, Physica A 303, 457 of view of the usual (extensive) approach the new equa- (2002) tions contain terms which can be formally identified with [21] I. Santamaria-Holek, R. F. Rodriguez, Physica A 366, terms appearing in the usual dissipative hydrodynamics 141 (2006) (d-hydrodynamics). Although this does not fully solve the [22] C. Tsallis, J. Stat. Phys. 52, 479 (1988) problems of d-hydrodynamics, nevertheless it allows us [23] C. Tsallis, Braz. J. Phys. 29, 1 (1999) to extend the usual perfect fluid approach (using only one [24] C. Tsallis, Physica A 340, 1 (2004) new parameter q) well beyond its usual limits, namely to- [25] C. Tsallis, Physica A 344, 718 (2004) ward the regions reserved so far only for the dissipative [26] C. Tsallis, Lect. Notes Phys. 560, 3 (2000) approach. [27] C. Tsallis, In: M. Gell-Mann, C. Tsallis (Eds.), Nonex- tensive Entropy - Interdisciplinary Applications (Ox- ford University Press, New York, 2004) Acknowledgements [28] C. Tsallis, AIP Conf. Proc. 965, 51 (2007) [29] G. Wilk, Z. Włodarczyk, Phys. Rev. Lett. 84, 2770 (2000) GW would like to express his gratitude towards the or- [30] T. S. Biró, A. Jakovác, Phys. Rev. Lett. 94, 132302 ganizers of the Int. Conf. on Statistical Physics - (2005) SigmaPhi2008 - Kolymbari, Crete, 14-18 July 2008 where [31] G. Wilk, Z. Włodarczyk, Chaos Soliton. Fract. 13, 581 this work was presented as an invited talk. Partial support (2001) (GW) from the Ministry of Science and Higher Education [32] O. V. Utyuzh, G. Wilk, Z. Włodarczyk, J. Phys. G 26, under contract 1P03B02230 is also acknowledged. L39 (2000)

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